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<h1 align=center> Functional equation of an $L$-function
</h1>
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The \begin{rawhtml}<a href="../4a">Riemann $\zeta$-function</a>\end{rawhtml} has functional equation
\begin{eqnarray}
\xi(s)&=&\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s) \cr
&=& \xi(1-s) .
\end{eqnarray}

\begin{rawhtml}<a href="../9a">Dirichlet $L$-functions</a>\end{rawhtml} satisfy the functional equation
\begin{eqnarray}
\xi(s,\chi)&=&\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2}+a)L(s,\chi) \cr
&=& **\xi(1-s,\overline{\chi}) ,
\end{eqnarray}
where $a=0$ if $\chi$ is even and $a=1$ if $\chi$ is odd,
and $*****$.

The \begin{rawhtml}<a href="../66a">Dedekind zeta function</a>\end{rawhtml} of a number field $K$
satisfies the functional equation
\begin{eqnarray}
\xi_K(s)&=&\left(\frac{\sqrt{|d_K|}}{2^{r_2}\pi^{n/2}}\right)^{s}
\Gamma(s/2)^{r_1}
\Gamma(s)^{r_2}\zeta_K(s)) \cr
&=& \xi_K(1-s).
\end{eqnarray}
Here $r_1$ and $2r_2$ are the number of real and complex conjugate embeddings
$K\subset \mathbb C$,
 $d_K$ is the discriminant,
and $n=[K,{\mathbb Q}]$ is the degree of $K/{\mathbb Q}$.
 
\begin{rawhtml}<a href="../13a">$L$-functions associated with a newform</a>\end{rawhtml} $f\in S_k(\Gamma_0(N)$
satisfy the functional equation
\begin{eqnarray}
\xi(s,f)&=&\left(\frac{\pi}{N}\right)^{-s}
\Gamma\left(s+\frac{k-1}{2}\right)L(s,f) \cr
&=& \varepsilon \xi(1-s,f) ,
\end{eqnarray}
where $a=0$ if $\chi$ is even and $a=1$ if $\chi$ is odd.

\begin{rawhtml}<a href="../14a">$L$-functions associated with a Maass newform</a>\end{rawhtml} 
with eigenvalue $\lambda=\frac14+R^2$ on $\Gamma_0(N)$
satisfy the functional equation
\begin{eqnarray}
\xi(s,f)&=&\left(\frac{N}{\pi}\right)^{s}
\Gamma\left(\frac{s+iR+a}{2}\right)\Gamma\left(\frac{s-iR+a}{2}\right)
L(s,f) \cr
&=& \varepsilon \xi(1-s,f) ,
\end{eqnarray}
where $a=0$ if $f$ is even and $a=1$ if $f$ is odd.

\begin{rawhtml}<a href="../44a">$GL(r)$ $L$-functions</a>\end{rawhtml} satisfy functional equations
of the form
$$
\Phi(s):= \left(\frac{N}{\pi^r}\right)^{s/2} 
\prod_{j=1}^r \Gamma\left(\frac{s+r_j}{2}\right) F(s) = 
\varepsilon \Phi(1-s).
$$

[This section needs a bit of work]\begin{rawhtml}
<p> <br>
<hr>
<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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